Bernal random loose packing through freeze-thaw cycling

We study the effect of freeze-thaw cycling on the packing fraction of equal spheres immersed in water. The water located between the grains experiences a dilatation during freezing and a contraction during melting. After several cycles, the packing fraction converges to a particular value ${\eta }_{\infty }=0.595$ independently of its initial value ${\eta }_0$. This behavior is well reproduced by numerical simulations. Moreover, the numerical results allow one to analyze the packing structural configuration. With a Voronoï partition analysis, we show that the piles are fully random during the whole process and are characterized by two parameters: the average Voronoï volume ${\mu }_v$ (related to the packing fraction $\eta$) and the standard deviation ${\sigma }_v$ of Voronoï volumes. The freeze-thaw driving modify the volume standard deviation ${\sigma }_v$ to converge to a particular disordered state with a packing fraction corresponding to the random loose packing fraction ${\eta }_{\text{BRLP}}$ obtained by Bernal during his pioneering experimental work. Therefore, freeze-thaw cycling is found to be a soft and spatially homogeneous driving method for disordered granular materials.

Figure 1

Sketch of the experimental setup. A high CCD camera records the top of the packing placed in a glass tube. The pile is back-illuminated by a homogeneous lighting system.

Figure 2

Top: Picture of the top of the pile (a) at the beginning of the cycle, (b) just before the freezing, (c) just after the freezing, (d) at the end of the freezing procedure, (e) during the heating, and (f) at the end of the cycle. Bottom: Temporal evolution of the temperature inside the pile, of the water height and of the granular height during one freeze-thaw cycle (${\eta }_0=0.573$).

Figure 3

Evolution of the packing fraction $\eta$ as a function of the freeze-thaw cycle number for experiments (top) and simulations (bottom). The squares and the circles correspond, respectively, to a loose and a close initial packing. The triangles correspond to an initial packing fraction close to the Bernal loose random packing fraction ${\eta }_{\text{BRLP}}=0.60$. The curves are fits from exponential law (see text).

Figure 4

Representation of the pile during one simulated freeze-thaw cycle. The grains are colored in yellow and the void spheres corresponding to the ice are colored in blue. From the left to the right: Initial pile with a low packing fraction, the void spheres are added, dilatation of the void spheres, and final state with a higher packing fraction.

Figure 5

Probability density function (PDF) of the Voronoï volumes $V$ for the initial piles (open symbols) and after 20 freeze-thaw cycling (plain symbols). The volumes $V$ are normalized with the volume average ${\mu }_v$ and the volume standard deviation ${\sigma }_v$. The conventions for the symbol shapes are the same as in Fig. 3. The data have been obtained from numerical simulations. The line corresponds to the gamma function.

Figure 6

Evolution of the Voronoï volume standard deviation ${\sigma }_v$ as a function of the packing fraction $\eta$ during 20 cycles for three different initial packing fractions. The data have been obtained from numerical simulations. The dashed curves are a guide for the eyes inspired from the work of Aste et al. [6].